Problem: Solve for $x$ : $ 2|x + 7| + 9 = 6|x + 7| + 6 $
Subtract $ {2|x + 7|} $ from both sides: $ \begin{eqnarray} 2|x + 7| + 9 &=& 6|x + 7| + 6 \\ \\ {- 2|x + 7|} && {- 2|x + 7|} \\ \\ 9 &=& 4|x + 7| + 6 \end{eqnarray} $ Subtract $6$ from both sides: $ \begin{eqnarray} 9 &=& 4|x + 7| + 6 \\ \\ {- 6} && {- 6} \\ \\ 3 &=& 4|x + 7| \end{eqnarray} $ Divide both sides by ${4}$ $ \dfrac{3} {{4}} = \dfrac{4|x + 7|} {{4}} $ Simplify: $ \dfrac{3}{4} = |x + 7| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -\dfrac{3}{4} = x + 7 $ or $ \dfrac{3}{4} = x + 7 $ Solve for the solution where $x + 7$ is negative: $ - \dfrac{3}{4} = x + 7$ Subtract ${7}$ from both sides: $ \begin{eqnarray} - \dfrac{3}{4} &=& x + 7 \\ \\ {- 7} && {- 7} \\ \\ -\dfrac{3}{4} - 7 &=& x \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $4$ $ - \dfrac{3}{4} {- \dfrac{28}{4}} = x $ $ -\dfrac{31}{4} = x $ Then calculate the solution where $x + 7$ is positive: $ \dfrac{3}{4} = x + 7 $ Subtract ${7}$ from both sides: $ \begin{eqnarray} \dfrac{3}{4} &=& x + 7 \\ \\ {- 7} && {- 7} \\ \\ \dfrac{3}{4} - 7 &=& x \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $4$ $ \dfrac{3}{4} {- \dfrac{28}{4}} = x $ $ -\dfrac{25}{4} = x $ Thus, the correct answer is $x = -\dfrac{31}{4} $ or $x = -\dfrac{25}{4} $.